 Solution

Apply the two-variable Itô formula to $e^w\cos b$. The independent Brownian motions satisfy $[B,W]=0$, while the two diagonal second derivatives sum to zero. ConsequentlyThus $Y$ is a continuous local martingale, but it is not Brownian motion. Brownian motion from any fixed starting point has quadratic variation $t$. If the displayed bracket were identically $t$, differentiating its continuous density would force $e^{2W_t}=1$ for every $t$, hence $W_t=0$ identically, an event of probability zero. This is the cosine-exponential Brownian local martingale. Its initial value $Y_0=1$ is retained in the next part.